How to apply Principal components (Eigenvectors) in PCA? I struggle to understand how to get further in my PCA analysis after computing the eigenvectors of the covariance matrix. 
I choose the number of principal components which contain 95% of the information and reduced the number of dimensions from 168 to 8. 
But now I end up with 8 eigenvectors. How do I use these? I don't know how to continue from this point. 
 A: Your data is $k=168$ dimensional, so each of your eigenvectors and your data vectors are $k$ dimensional. Eigenvectors represent directions of change in your data. They're like your new x,y,z axes. When your select $m=8$ of them, it means you select $m$ of these axes. Then, the data is projected onto these axes. The projection is done via dot products. For example, if the data vector is $d=[2,1,3]$ and we project this vector onto $x$ axis (i.e. $\hat{x}=[1,0,0]$), we calculate the dot product $<\hat{x},d>=2$, and get the $x$ coordinate of your data vector $d$. By projecting the data onto new axes, i.e. onto the eigenvectors, we get our new coordinates. So, each of your $k$ dimensional original data points will map to $m$ dimensional new data points.
A: Then you multiply the matrix of original features (let’s say $X$) by the matrix whose columns are the k chosen eigenvectors (let’s say $V$) corresponding to the k largest eigenvalues (the 8 representing 95% of the total variance) to get to the reduced-set of transformed uncorrelated features:
$$X^{*}=XV$$
Which is a matrix with 8 columns and a number of rows equal to the number of samples that you have, as this matrix represents the 8 PCA-transformed features observed  for each sample (row of $X^{*} $). Now you can use this new set of features for multiple purposes, including (which is very common!) the estimation a model for the prediction of a certain dependent variable, with the desiderabile properties that, compared to the previous features, the new transformed ones are not correlated and they have reduced the number of predictors from 168 to 8 while retaining a large percentage (95%) of their original total variance.
